Mass Transport at Infinite Regular Arrays of Microband Electrodes

Article pubs.acs.org/ac

Mass Transport at Infinite Regular Arrays of Microband Electrodes Submitted to Natural Convection: Theory and Experiments Cécile Pebay, Catherine Sella, Laurent Thouin,* and Christian Amatore Ecole Normale Supérieure, Département de Chimie, UMR CNRS-ENS-UPMC 8640 Pasteur, 24 rue Lhomond, 75231 Paris Cedex 05, France ABSTRACT: Mass transport at infinite regular arrays of microband electrodes was investigated theoretically and experimentally in unstirred solutions. Even in the absence of forced hydrodynamics, natural convection limits the convectionfree domain up to which diffusion layers may expand. Hence, several regimes of mass transport may take place according to the electrode size, gap between electrodes, time scale of the experiment, and amplitude of natural convection. They were identified through simulation by establishing zone diagrams that allowed all relative contributions to mass transport to be delineated. Dynamic and steady-state regimes were compared to those achieved at single microband electrodes. These results were validated experimentally by monitoring the chronoamperometric responses of arrays with different ratios of electrode width to gap distance and by mapping steady-state concentration profiles above their surface through scanning electrochemical microscopy.

E

microdisk arrays,27,28 and microband arrays11,18,29 as well as to nanoelectrodes5,8,9,19,30 and nano-microparticles31−34 through consideration of pure diffusional mass transport. The purpose of this study is to investigate the influence of natural convection on mass transport at infinite regular arrays of microband electrodes. All microbands are parallel and uniform and are arranged periodically, meaning that they have identical width and length. They are separated from their neighbors by an insulating material in which they are embedded. The microband length is long enough so that the problem can be considered in two dimensions. Furthermore, the arrays are composed of a sufficiently large number of electrodes so that edge effects at both ends of the arrays are negligible. This situation applies to all geometries of arrays whose electrochemical behaviors remain independent of the number of electrodes. In previous studies, we have demonstrated the importance of natural convection, which alters diffusional responses predicted for electrochemical properties of planar electrodes35−37 and single microdisk38,39 and microband electrodes40 at long experimental cases, i.e., when steady-state regimes are achieved.41,42 The situation thus becomes complicated when overlapping of diffusion layers is considered between microbands. In the following, we delineate theoretically and experimentally the conditions under which natural convection preempts the validity of theoretical18,43 and/or experimental11,18 interpretations based on pure diffusion control at infinite microband arrays. According to the geometrical parameters of the array and time scale of the experiment, numerical simulations were performed by

lectrodes having at least one dimension smaller than a few tens of micrometers have several advantages over electrodes of larger dimensions. Low background currents, rapid time responses due to fast double-layer charging, and steady-state currents are the main benefits resulting from their small sizes.1−3 As mass transport to the electrode surface is enhanced, high current densities prevail because of the involvement of nonlinear diffusion and edge effects. In turn, their small surface areas lead to the measurement of low current intensities. Therefore, the use of arrays or ensembles of micro- and nanoelectrodes provides a good opportunity to keep the exceptional properties of small electrodes while measuring higher total current output.4−10 Furthermore, such arrays offer additional and significant advantages over solid electrodes of the same surface area. Because the signal-to-noise ratio is enhanced, the analytical detection limits and sensitivities per unit active area are further improved.2,10 The response of the array depends on the relative sizes of the active and insulating elements with respect to the thickness of the diffusion layers that develop above their surface. For most electroanalytical applications and best optimal performances,4,7,11−14 the array should be designed so that the distance between two adjacent electrodes is large enough to avoid any overlapping of diffusion layers during the time scale of the experiment but not too large to avoid any waste of the surface area.15,16 The dynamics of mass transport taking place at such electrode arrays have already been described.17−19 Several behaviors or kinetic categories may be defined in relation to linear and nonlinear diffusion and overlapping of diffusion layers at individual electrodes. These categories were initially defined to illustrate the voltammetric behaviors of a simple process at partially blocked electrodes.20,21 They have been successfully implemented to microdisk arrays,9,22−27 recessed © 2013 American Chemical Society

Received: October 2, 2013 Accepted: November 24, 2013 Published: November 27, 2013 12062

dx.doi.org/10.1021/ac403159j | Anal. Chem. 2013, 85, 12062−12069

Analytical Chemistry

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concentrations at any location in the solution follow the balance

implementing a known and well-validated model of natural convection.36 In order to evaluate the performance of the arrays, mass-transport regimes were delineated according to dimensionless parameters. These results were then compared to experimental chronoamperometric responses and concentration profiles obtained by scanning electrochemical microscopy (SECM) at arrays of different geometries.

cA + c B = c 0

where c is the initial concentration of the substrate A in bulk solution. Because the length of microbands l is much larger than the width w, the diffusional contribution at both of their ends is negligible compared to the main surface of the microbands. The problem then reduces to a two-dimensional (2D) problem, which can be solved by resolution of the following masstransport equation upon considering pure diffusion:

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EXPERIMENTAL SECTION All of the solutions were prepared in purified water (Milli-Q, Millipore) using 0.1 M KNO3 (Fluka) as the supporting electrolyte. Ferrocenylmethanol, FcCH2OH (Acros), was used as the redox probe at 1 mM concentration. For both species FcCH2OH and FcCH2OH+, the diffusion coefficient was assumed to be equal to D = (7.6 ± 0.5) × 10−6 cm2 s−1.40 The thickness of the convection-free layer δconv was determined independently before each experiment by chronoamperometry using a method described previously.36 Three configurations of microband arrays were designed with three different gaps g between electrodes having nominal values of 20, 80, and 180 μm (with effective values based on optical microscopy measurements 22.8, 81.7, and 180.5 μm, respectively). Each array was composed of 51 regular parallel platinum microbands of several millimeters length and width w equal to 19 μm. They were patterned on glass substrates by classical soft lithography and lift-off procedures.44 The counter electrode was a platinum coil. A large platinum band electrode located on the glass substrate far from the array was used as a pseudoreference electrode (REF). A scanning electrochemical microscope (910B, CH Instruments) was used to measure the steady-state concentration profiles developing above each array. The tip consisted of a platinum disk electrode of submicrometric dimension (∼380 nm radius) with a glass diameter to electrode diameter RG equal to 13. Its fabrication and procedure used to map the local concentrations have already been reported elsewhere.45 Under these conditions, the apparent potential of the FcCH2OH/ FcCH2OH+ redox couple was estimated at +0.1 V/REF. All of the electrodes in the array were biased on the oxidation plateau of FcCH2OH at +0.25 V/REF. The tip was biased respectively at +0.25 V/REF to collect FcCH2OH or at −0.1 V/REF to detect FcCH2OH+. A scan rate lower than 20 μm s−1 was used for the tip displacement. Under these experimental conditions, it was checked that convective effects generated by the displacement were negligible with no apparent distortion of the concentration profiles. For this purpose, the experimental measurements were compared to those obtained at smaller displacement rates (data not shown). As described previously,40 the mass-transport equation was solved numerically by finite elements using commercial software (Comsol Multiphysics v.4.2).

⎛ ∂ 2c ∂c ∂ 2c ⎞ = D∇2 c = D⎜ 2 + 2 ⎟ ∂t ∂y ⎠ ⎝ ∂x

(3)

∇ is the Laplacian in Cartesian coordinates x and y. Previously, we demonstrated that the influence of natural convection can be taken into account in the mass-transport equation by introducing an apparent diffusion coefficient Dapp depending on the orthogonal distance y from the electrode surface:44 2

⎡ ⎛ y ⎞4 ⎤ Dapp = D⎢1 + γ ⎜ ⎟⎥ ⎢⎣ ⎝ δconv ⎠ ⎥⎦

(4)

where γ is a constant equal to 1.522 and δconv is the thickness of the convection-free domain over the electrode surface. δconv is the only parameter in the model accounting for the influence of natural convection on mass transport, and it may be determined independently from the steady-state current observed on a millimetric electrode.44 To simplify resolution of the problem, dimensionless parameters were introduced for concentration

C = c /c 0

(5)

coordinates X = 2x /w

and

Y = 2y/w

(6)

geometry of the array or gap distance between electrodes G = 2g /w

(7)

thickness of the convection-free layer Δconv = 2δconv /w

(8)

time

τ = 4Dt /w 2

(9)

Combinations of eqs 3−6 and eqs 8 and 9 lead thus to a new mass-transport equation:

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⎡ ⎛ Y ⎞4 ⎤ ∂C ⎢ = ∇ 1 + γ⎜ ⎟ ∇C ⎥ ⎢⎣ ⎥⎦ ∂τ ⎝ Δconv ⎠

THEORY AND SIMULATIONS All microband electrodes in the array have identical width w and length l. Each is separated from its neighbors by an identical gap of width g. At the interface, the electrochemical reaction is a one-electron, fast, and chemically reversible process: A ⇄ B + e−

(2)

0

(10)

The semi-infinite space used for simulation is now described by the coordinates (X, Y). Because the number of elements (i.e., electrodes + insulating material) in the array is supposed to be infinite (as it is closely approached experimentally in this work, see below), the problem can be solved by considering only a single unit cell of the array.20 Indeed, according to the symmetry of the system, the array can be divided into unit cells

(1)

The diffusion coefficients of both the substrate and product are assumed to be identical and equal to D. Hence, the A and B 12063



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where F is the Faraday.

comprised between two parallel planes located respectively along the central axis of a band and at the midpoint between two adjacent electrodes (Figure 1). On the X axis, the unit cell is delimited between X = 0 and X = 1 + G/2.

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RESULTS AND DISCUSSION Diffusive and Convective Regimes. Figure 2 shows an example of simulated diffusion layers that may expand over an

Figure 1. Cell unit (in gray area) used for numerical simulations in a 2D dimensionless space (X, Y). G is the gap distance between two adjacent electrodes. Electrodes are indicated by black rectangles.

Inside the cell, the mass-transport equation (eq 10) was solved numerically in association with the boundary conditions that apply to chronoamperometry: initial condition

C=1

τ = 0,

(11)

electrode surface 0 ≤ X ≤ 1,

τ > 0,

Y = 0,

C=0

(12)

semi-infinite boundary Y → ∞,

τ ≥ 0,

C=1

Figure 2. Dimensionless concentration profiles simulated during chronoamperometry at an infinite microband array in the absence (left column, Δconv → ∞) and presence (right column, Δconv = 26) of natural convection for several times τ. In each case, two half-electrodes are represented. The indicated regimes refer to those defined in Table 1. G = 2.4.

(13)

insulating boundary 1 0,

Y > 0,

∂C /∂X = 0

array with G = 2.4. In each case, two unit cells were considered in order to clearly illustrate concentration profiles in the absence (left column) and presence (right column) of natural convection. At very short time, the thickness of the diffusion layers is small compared to the electrode width, gap distance, and amplitude Δconv of the free convection layer. Mass transport to each individual electrode is mainly controlled by planar diffusion regardless of array geometry G and Δconv (Figure 2, τ = 0.01). At much longer time, planar diffusion is still observed in both situations (Figure 2, τ = 900). Indeed, diffusion layers between adjacent electrodes fully overlap, leading to a planar diffusion field prevailing over the whole surface of the array including the electrodes and insulating elements. This situation arises at this time scale because the gap distance between electrodes is small enough compared to the thickness of the diffusion layers. However, even if the situation is similar in the absence or presence of natural convection, the characteristics of the two regimes are quite different. Without natural convection, diffusion layers still expand with time to greater distances like for a Cottrellian-type behavior. In contrast, when natural convection starts to interfere with mass transport, it prevents the development of diffusion layers beyond a limiting thickness equal to Δconv. In this case, a steadystate regime controlled by planar diffusion is established over the whole surface of the array (Figure 2, τ = 900).

(15)

insulating boundary G , Y > 0, ∂C /∂X = 0 (16) 2 According to the time scale of the experiment, the finite dimension Ycell of the cell unit was checked to be sufficiently large and at least superior to 6√τ in order to fulfill the semiinfinite boundary condition (eq 13) on the Y axis. The dimensionless current Ψ of an individual electrode in the array is evaluated by integrating the flux over half of the electrode width according to X=1+

τ > 0,

Ψ=2

∫0

1 ⎛ ∂C ⎞ ⎜

⎟

⎝ ∂Y ⎠Y = 0

dX (17)

Because edge effects on both sides of the array are not considered and neglected because of a large number of elements, the global current delivered by the N electrodes of the array is Ψarray = N Ψ

(18)

The corresponding current iarray is thus given by iarray = NFlDc°Ψ

(19) 12064



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surface of the unit cell including electrodes and insulating elements), the current then follows (Figure 3A)

These two situations perfectly illustrate the difficulty of the problem when one really wants to describe the dynamic of mass transport according to variations in G, Δconv, and time scale τ of the experiment. This is particularly the case if Δconv is unknown or its influence neglected. In addition, other regimes may be encountered at intermediate times such as in Figure 2 for τ = 0.25 and 10 or for other G values, when overlapping between diffusion layers is more or less intense. All of these behaviors can be observed and well characterized from the time variations of currents, as shown in Figure 3A. As

Ψ=

2+G πτ

(21)

Conversely, when the range of time scale considered is such that natural convection is not negligible, a steady-state current is reached. On the basis of the Nernst’s layer concept and by considering Δconv as the limiting thickness of the diffusion layer, the current is given by Ψ=

2+G Δconv

(22)

Note that both limiting situations (eqs 21 and 22) are different from those expected in the case of a single electrode (Figure 3B).40 Indeed, at long enough time scale, a quasi-steady-state regime is achieved at single band electrodes46 because of the development of a hemicylindrical-type diffusion: 2π Ψ= ln(16τ ) (23) or a steady-state regime is installed when diffusion is limited because of natural convection: 2 Ψ= Δconv (24) To quantify the prevalence of each contribution or regime (eqs 20−24) with mass transport, we simulated the current responses over a wider range of conditions. Actually, this approach implements our previous study on single microband electrodes40 by introducing the parameter G in association with τ and Δconv. Boundaries from one regime to the other were estimated by setting a relative threshold on Ψ using the criterion

Ψref − Ψ = 0.1 Ψref

(25)

where Ψref refers to one of the five limiting behaviors identified above. Equations of Ψref are summarized in Table 1, where a Figure 3. Dimensionless current simulated during chronoamperometry at infinite microband arrays (A) and at single microbands (B) in the absence (Δconv → ∞, solid circles) and presence (Δconv = 26, open circles) of natural convection: (A) for G = 2.4, the Pw+g regime as the dotted line (eq 21) and the Cw+g regime as the solid line (eq 22); (B) the Hw regime as the dotted line (eq 23) and the Cw regime as the solid line (eq24); (A and B) the Pw regime as the dashed line (eq 20).

Table 1. Equations of Mass-Transport Regimes and Criteria Based on Equation 25 for Delineating Regimes Controlled by Planar Diffusion, Hemicylindrical Diffusion, and Natural Convection over Infinite Microband Arraysa single electrode behaviors

long as the gap distance separating two adjacent microbands is sufficiently large to prevent any interaction between diffusion layers, electrodes behave like they were isolated and independent. Their responses are the same as those of single electrodes of identical dimension (see Figure 3B). At very short times, the current follows the well-known Cottrell equation: 2 Ψ= (20) πτ

planar diffusion hemicylindrical diffusion natural convection

other regimes

regimes

Ψref

regimes

Ψref

Pw Hw Cw

eq 20 eq 23 eq 24

Pw+g

eq 21

Cw+g

eq 22

a

A distinction is made between regimes resulting from single electrode behaviors and regimes only observed at infinite arrays.

distinction is specifically made between regimes issued from single electrode behaviors (Pw, Cw, and Hw) and regimes observed only at arrays (Pw+g and Cw+g). Note that, because arrays are supposed to be of infinite size, hemicylindrical diffusion over their entire surface cannot take place. This situation, which applies in the case of a limited number of electrodes N, will not be considered in the following.

At longer times, diffusion layers start to overlap. The current clearly deviates from eq 20 and tends toward other limiting situations. When natural convection does not interfere, a Cottrellian behavior is again observed after a transition period. By considering the whole surface of the array (i.e., the whole 12065



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Zone Diagrams. Zone diagrams were thus established based on Ψ criteria (eq 25 and Table 1) to delineate the contributions of all possible regimes identified at infinite arrays. In Figure 4, the parameters τ and Δconv were used as

According to the geometry G of the arrays, three major situations may be distinguished: (i) For large G, the electrodes behave as isolated electrodes (Figure 4A). The zone diagram is identical with that previously established in the case of single electrodes.40 (ii) For intermediate G values, the diffusion fields of adjacent electrodes overlap sufficiently so that the electrode currents deviate from that of an isolated electrode (Figure 4B,C). Regimes Pw+g and Cw+g enlarge progressively upon decreasing G in detriment to regimes Hw and Iconv, which cancel out as soon as G < 1. Note that the overlapping of diffusion layers in conjunction with the effect of natural convection can preempt considerably the conditions under which hemicylindrical diffusion at individual electrodes (Hw regime) is usually expected (Figure 4B). (iii) For very small G, there is a complete and strong overlap of diffusion fields (Figure 4D). The current at each electrode is almost equal to that expected for linear diffusion at a single electrode of the same dimension. Regime Pw+g merges with regime Pw. The same situation applies for Cw+g with Cw. As an illustration, we reported in Figure 2 the regimes corresponding to the concentration profiles simulated for G = 2.4. With no influence of natural convection, a sequence of regimes from Pw to Idiff to Pw+g is observed as a function of time. In comparison, steady-state regime Cw+g is further reached in the presence of natural convection, demonstrating its great influence at a long time scale. To easily predict steady-state regimes Cw, Cw+g, and Iconv, we thus established a simple relation to fit in Figure 4 the criterion based on Ψdiff. This led to the equivalence between Δconv and a diffusion length:

Figure 4. Zone diagrams (Δconv and τ) established from simulated currents at infinite microband arrays for several geometries G: (A) G > 400; (B) G = 8; (C) G = 2; (D) G < 0.4. Regimes refer to those defined in Tables 1 and 2. The dashed lines represent the boundary issued from the Ψdiff criterion.

Δconv ≈

coordinates to differentiate conditions under which each regime prevails. Another condition was also considered to be able to distinguish the domain where regimes are exclusively controlled by diffusion (i.e., without any contribution of natural convection). This was performed by evaluating the current Ψdiff when Δconv is infinite and has no influence on Ψ (Table 2).

diffusion natural convection single electrode behavior a

conditions

regimes

Ψdiff Ψconv Ψs

Δconv → ∞ Δ = Δconv G→∞

Pw, Idiff, Hw, Pw+g Cw, Cw+g, Iconv Pw, Idiff, Hw, Cw, Iconv

(26)

This relation is a numerical approximation that encompasses almost all conditions reported in Figure 4. Its accuracy is higher when the Hw regime cancels out for G ≤ 2 (see Figure 4C). Therefore, when Δconv is higher than (πτ)1/2, regimes are convection - independent. In the opposite case, they obey a thin-layer-type regime because of natural convection. Figure 5 shows two other zone diagrams corresponding to both limiting situations described above. The first one [Figure 5A; Δconv > (πτ)1/2] describes all conditions required to observe diffusion regimes according to parameters G and τ. It is reminiscent to diagrams already established for microdisk27 and microband18 arrays but with different criteria and coordinates. The second diagram [Figure 5B; Δconv < (πτ)1/2] shows all steady-state regimes established under natural convection as a function of G and Δconv. To implement the diagrams, we also reported another condition related to single electrode behavior. It was established via eq 25 by considering the current Ψs evaluated when the gap distance does not contribute to Ψ (Table 2). This boundary is very convenient because it formally delineates optimal conditions for using infinite arrays. Indeed, the gap distance between electrodes needs to be large enough to exclude any overlay between diffusion layers while keeping the properties of single electrodes. Under diffusion control, it can be observed that these conditions correspond to regimes Pw, Idiff, and Hw (Figure 5A). Under the influence of natural convection, these conditions apply also to regime Cw and partially to regime Iconv (Figure 5B). However, one must underline that two other regimes, Pw+g (Figure 5A) and Cw+g (Figure 5B), may present also some advantages. Indeed, under these specific conditions, the current is proportional to w + g, while the noise resulting from the capacitive contribution is proportional to w. Compared to a

Table 2. Additional Criteria Based on Equation 25 for Delineating Regimes Controlled by Diffusion, Natural Convection, or Regimes Resulting from Single Electrode Behaviorsa Ψref

πτ

Δ is equal to 2δ/w, where δ is the thickness of the diffusion layers.

As expected, this condition based on the Ψdiff criterion delimits in the upper part of each diagram a wide domain that encompasses regimes Pw, Hw, and Pw+g (Figure 4A−C). In particular, it allows intermediate regimes Idiff to be delimited from another one, Iconv. One must stress that this boundary is also valid to set the limits of steady-state regimes Cw and Cw+g resulting from natural convection. Indeed, similar results were found between the Ψdiff criterion and the reciprocal criterion based on Ψconv (see Table 2 and the related definition of Ψconv). No real distinction could be made graphically because the transitions between regimes due to diffusion and convection were very sharp. For the sake of simplicity, this latter boundary was not reported in the diagrams. 12066



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Figure 6. Comparison between experimental (circles) and simulated (curves) currents at microband arrays having several gap distances g. Currents simulated in the absence (dashed line, δconv → ∞) and presence (solid curves, δconv = 250 μm) of natural convection. w = 19 μm and N = 51. FcCH2OH/0.1 M KNO3.

In this investigation, mapping of the concentration profiles by SCEM illustrated also perfectly the drastic influence of natural convection above infinite arrays and overlapping between diffusion layers (Figure 7). On the one hand, concentration gradients extended in the y direction over a shorter distance than expected for pure diffusion because of the finite extent of the convection-free layer thickness δconv. On the other hand, a decrease in the gap distance between electrodes leads to a drastic homogenization of the solution above the surface of arrays. A very good agreement was also noticed between experimental and theoretical data, but only when natural convection was accounted for. These results in Figures 6 and 7 thus validate the predictions issued from the present theory.

Figure 5. Zone diagrams established from simulated currents at infinite microband arrays under limiting conditions. (A) Dynamic regimes under diffusion control with Δconv > (πτ)1/2. (B) Steady-state regimes under the influence of natural convection with Δconv < (πτ)1/2. In each case, the dashed line represents the boundary issued from the Ψs criterion.

large single electrode of equivalent surface w + g, arrays are thus expected to provide a better signal-to-noise ratio. Experimental Illustrations. Figure 6 shows experiments that were performed in chronoamperometry at microband arrays and compared to a single microband electrode of the same dimension. It was checked by complete simulation of the array that the configurations used (w, g, N) corresponded to the case of infinite arrays with no influence on the current of edge effects on both sides of the arrays (data not shown here). In each experiment, the data were systematically compared to predictions based on simulations performed in the absence and presence of natural convection. δconv was estimated to be experimentally equal to 250 μm. As expected, the current of the arrays decreases with the gap between electrodes. Indeed, the overlap between diffusion layers starts to proceed earlier when the gap is smaller. In all cases, steady-state currents are monitored at long time because of natural convection. These effects were in full agreement with those predicted in the presence and absence of natural convection. All regimes identified in Tables 1 and 2 were covered by these experimental conditions except regimes Cw and Pw+g. Application of eq 26 gave a time of about 26 s, which corresponds indeed to the time above which the currents become altered by natural convection. As can be observed, a good agreement was obtained between simulated and experimental data in all cases.

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CONCLUSION This study evidenced and characterized all mass-transport regimes that may occur at infinite arrays of microband electrodes. In comparison to previous theories, it has been extended by considering the drastic influence of natural convection on mass transport. Several dynamic and steadystate regimes were clearly identified by applying a wellestablished model for the treatment of natural convection. A comparison with single electrode behaviors showed that overlapping of diffusion layers in conjunction with the influence of natural convection may preempt severely the expected performances of band arrays when considering pure diffusion regimes. Indeed, the hemicylindrical regimes that are usually exploited at an individual band electrode may only be available over a limited range of experimental conditions. The excellent agreement found between predictions and experimental data validated the present theory and added a further confirmation of our treatment of natural convection. Zone diagrams were constructed to delineate all regimes according to dimensionless parameters, hence providing all of the necessary conditions to 12067



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Figure 7. Comparison between the experimental (symbols) and simulated (curves) concentration profiles established at several distances y from microband arrays presenting different gap distances g. Concentration profiles simulated in the absence (dashed line, δconv → ∞) and presence (solid curves, δconv = 250 μm) of natural convection. The electrode positions are materialized by black rectangles on the x axis. w = 19 μm and N = 51. FcCH2OH/0.1 M KNO3. (10) Henstridge, M. C.; Compton, R. G. Chem. Rec. 2012, 12, 63−71. (11) Le Drogoff, B.; El Khakani, M. A.; Silva, P. R. M.; Chaker, M.; Vijh, A. K. Electroanalysis 2001, 13, 1491−1496. (12) Compton, R. G.; Cutress, I. J. Electroanalysis 2009, 21, 2617− 2625. (13) Huang, X.-J.; O’Mahony, A. M.; Compton, R. G. Small 2009, 5, 776−788. (14) Orozco, J.; Fernández-Sánchez, C.; Jiménez-Jorquera, C. Sensors 2010, 10, 475−490. (15) Chevallier, F. G.; Compton, R. G. Electroanalysis 2006, 18, 2369−2374. (16) Menshykau, D.; Huang, X. J.; Rees, N. V.; del Campo, F. J.; Munoz, F. X.; Compton, R. G. Analyst 2009, 134, 343−348. (17) Davies, T. J.; Banks, C. E.; Compton, R. G. J. Solid State Electrochem. 2005, 9, 797−808. (18) Streeter, I.; Fietkau, N.; Del Campo, J.; Mas, R.; Munoz, F. X.; Compton, R. G. J. Phys. Chem. C 2007, 111, 12058−12066. (19) Godino, N.; Borrise, X.; Munoz, F. X.; del Campo, F. J.; Compton, R. G. J. Phys. Chem. C 2009, 113, 11119−11125. (20) Amatore, C.; Saveant, J. M.; Tessier, D. J. Electroanal. Chem. 1983, 147, 39−51. (21) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1984, 160, 19−26. (22) Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M. J. Electroanal. Chem. 1987, 217, 417. (23) Scharifker, B. R. J. Electroanal. Chem. 1988, 240, 61−76. (24) Lee, H. J.; Beriet, C.; Ferrigno, R.; Girault, H. H. J. Electroanal. Chem. 2001, 502, 138−145. (25) Davies, T. J.; Ward-Jones, S.; Banks, C. E.; del Campo, J.; Mas, R.; Munoz, F. X.; Compton, R. G. J. Electroanal. Chem. 2005, 585, 51− 62. (26) Ordeig, O.; Banks, C. E.; Davies, T. J.; del Campo, J.; Mas, R.; Munoz, F. X.; Compton, R. G. Analyst 2006, 131, 440−445. (27) Guo, J.; Lindner, E. Anal. Chem. 2009, 81, 130−138.

optimize the sought properties of microband arrays under the influence of natural convection whatever their geometries.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported in part by the CNRS (Grant UMR8640), Ecole Normale Supérieure, UPMC, and French Ministry of Research.

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Mass Transport at Infinite Regular Arrays of Microband Electrodes

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